Abelian-extensible automorphism-invariant subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

Suppose $G$ is an abelian group and $H$ is a subgroup of $G$. We say that $H$ is an abelian-extensible automorphism-invariant subgroup of $G$ if, for every abelian-extensible automorphism $\sigma$ of $G$, we have $\sigma(H) = H$.

Formalisms

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression $H$ is a fully invariant subgroup of $G$ if ... This means that full invariance is ... Additional comments
abelian-extensible automorphism $\to$ function every abelian-extensible automorphism of $G$ sends every element of $H$ to within $H$ the invariance property for abelian-extensible automorphisms
abelian-extensible automorphism $\to$ endomorphism every abelian-extensible automorphism of $G$ restricts to an endomorphism of $H$ the endo-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a property stronger than the property of being an endomorphism
abelian-extensible automorphism $\to$ automorphism every abelian-extensible automorphism of $G$ restricts to an automorphism of $H$ the auto-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a group-closed property of automorphisms

Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Characteristic subgroup of abelian group Abelian-potentially characteristic subgroup|FULL LIST, MORE INFO
Abelian-potentially characteristic subgroup a characteristic subgroup in some bigger abelian group abelian-potentially characteristic implies abelian-extensible automorphism-invariant |FULL LIST, MORE INFO
Subgroup of finite abelian group subgroup of finite abelian group (via abelian-potentially characteristic; see finite abelian NPC theorem) Abelian-potentially characteristic subgroup|FULL LIST, MORE INFO

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Subgroup of abelian group subgroup of abelian group not implies abelian-extensible automorphism-invariant